Earned Value Management derives a variety of information from a Gantt chart. (assuming that the Gantt chart has been built to support the analysis).
Critical path(s) (@Stephan pointed this out and deserves the credit.) Calculation of critical path is vital; automated analysis could concentrate on multiple critical paths.
Monte Carlo Analysis - this is probably the most valuable thing you could add. Monte Carlo Analysis is not easy to set up, but is the only way to answer questions about the consequences of multiple risks/opportunities.
If you are at a Project Management Office, and you have access to multiple Gantt charts from multiple projects, it might be interesting to analyze the charts for:
Project closure rate - how many projects close successfully and how many close before the value is delivered?
Estimation accuracy/variance - how close is estimated duration to actual duration? What variables affect accuracy. You'll need to separate issues of chance (aleatory issues like "the supplier had a trucking strike and was late in delivering materials", from epistemelogical issues like, "The dev team overestimated their familiarity with the BlueDuck programming framework."
Which types of work packages were late/early and by how much?
WHich teams were late/early? By how much and why?
If you have access to a set of Gantt charts, you could probably learn a lot by studying change mangement across the set.
- What is the velocity of change? (A colleague once worked on a project that had more than 100 authorized changes in a quarter; this was a clear signal of problems with the project. The project I was managing at the time had changes only prior to financial reviews - which was a sign of a different kind of problem).